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1. Stochastic Lagrangian dynamics of vorticity. Part 1. General theory for viscous, incompressible fluids

   https://doi.org/10.1017/jfm.2020.491  

   Eyink, Gregory L. ; Gupta, Akshat ; Zaki, Tamer A. ( October 2020 , Journal of Fluid Mechanics)

   null (Ed.)

   Prior mathematical work of Constantin & Iyer ( Commun. Pure Appl. Maths , vol. 61, 2008, pp. 330–345; Ann. Appl. Probab. , vol. 21, 2011, pp. 1466–1492) has shown that incompressible Navier–Stokes solutions possess infinitely many stochastic Lagrangian conservation laws for vorticity, backward in time, which generalize the invariants of Cauchy ( Sciences mathématiques et physique , vol. I, 1815, pp. 33–73) for smooth Euler solutions. We reformulate this theory for the case of wall-bounded flows by appealing to the Kuz'min ( Phys. Lett. A , vol. 96, 1983, pp. 88–90)–Oseledets ( Russ. Math. Surv. , vol. 44, 1989, p. 210) representation of Navier–Stokes dynamics, in terms of the vortex-momentum density associated to a continuous distribution of infinitesimal vortex rings. The Constantin–Iyer theory provides an exact representation for vorticity at any interior point as an average over stochastic vorticity contributions transported from the wall. We point out relations of this Lagrangian formulation with the Eulerian theory of Lighthill (Boundary layer theory. In Laminar Boundary Layers (ed. L. Rosenhead), 1963, pp. 46–113)–Morton ( Geophys. Astrophys. Fluid Dyn. , vol. 28, 1984, pp. 277–308) for vorticity generation at solid walls, and also with a statistical result of Taylor ( Proc. R. Soc. Lond. A , vol. 135, 1932, pp. 685–702)–Huggins ( J. Low Temp. Phys. , vol. 96, 1994, pp. 317–346), which connects dissipative drag with organized cross-stream motion of vorticity and which is closely analogous to the ‘Josephson–Anderson relation’ for quantum superfluids. We elaborate a Monte Carlo numerical Lagrangian scheme to calculate the stochastic Cauchy invariants and their statistics, given the Eulerian space–time velocity field. The method is validated using an online database of a turbulent channel-flow simulation (Graham et al. , J. Turbul. , vol. 17, 2016, pp. 181–215), where conservation of the mean Cauchy invariant is verified for two selected buffer-layer events corresponding to an ‘ejection’ and a ‘sweep’. The variances of the stochastic Cauchy invariants grow exponentially backward in time, however, revealing Lagrangian chaos of the stochastic trajectories undergoing both fluid advection and viscous diffusion. 

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2. Large deviations for small noise diffusions over long time

   https://doi.org/10.1090/btran/172  

   Budhiraja, Amarjit ; Zoubouloglou, Pavlos ( January 2024 , Transactions of the American Mathematical Society, Series B)

   We study two problems. First, we consider the large deviation behavior of empirical measures of certain diffusion processes as, simultaneously, the time horizon becomes large and noise becomes vanishingly small. The law of large numbers (LLN) of the empirical measure in this asymptotic regime is given by the unique equilibrium of the noiseless dynamics. Due to degeneracy of the noise in the limit, the methods of Donsker and Varadhan [Comm. Pure Appl. Math. 29 (1976), pp. 389–461] are not directly applicable and new ideas are needed. Second, we study a system of slow-fast diffusions where both the slow and the fast components have vanishing noise on their natural time scales. This time the LLN is governed by a degenerate averaging principle in which local equilibria of the noiseless system obtained from the fast dynamics describe the asymptotic evolution of the slow component. We establish a large deviation principle that describes probabilities of divergence from this behavior. On the one hand our methods require stronger assumptions than the nondegenerate settings, while on the other hand the rate functions take simple and explicit forms that have striking differences from their nondegenerate counterparts.

    

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3. Sharp bounds for decomposing graphs into edges and triangles

   https://doi.org/10.1017/S0963548320000358  

   Blumenthal, Adam ; Lidický, Bernard ; Pehova, Yanitsa ; Pfender, Florian ; Pikhurko, Oleg ; Volec, Jan ( March 2021 , Combinatorics, Probability and Computing)

   null (Ed.)

   Abstract For a real constant α , let $\pi _3^\alpha (G)$ be the minimum of twice the number of K 2 ’s plus α times the number of K 3 ’s over all edge decompositions of G into copies of K 2 and K 3 , where K r denotes the complete graph on r vertices. Let $\pi _3^\alpha (n)$ be the maximum of $\pi _3^\alpha (G)$ over all graphs G with n vertices. The extremal function $\pi _3^3(n)$ was first studied by Győri and Tuza ( Studia Sci. Math. Hungar. 22 (1987) 315–320). In recent progress on this problem, Král’, Lidický, Martins and Pehova ( Combin. Probab. Comput. 28 (2019) 465–472) proved via flag algebras that $\pi _3^3(n) \le (1/2 + o(1)){n^2}$ . We extend their result by determining the exact value of $\pi _3^\alpha (n)$ and the set of extremal graphs for all α and sufficiently large n . In particular, we show for α = 3 that K n and the complete bipartite graph ${K_{\lfloor n/2 \rfloor,\lceil n/2 \rceil }}$ are the only possible extremal examples for large n . 

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4. Tempered fractional LES modeling

   https://doi.org/10.1017/jfm.2021.955  

   Samiee, Mehdi ; Akhavan-Safaei, Ali ; Zayernouri, Mohsen ( February 2022 , Journal of Fluid Mechanics)

   The presence of non-local interactions and intermittent signals in the homogeneous isotropic turbulence grant multi-point statistical functions a key role in formulating a new generation of large-eddy simulation (LES) models of higher fidelity. We establish a tempered fractional-order modelling framework for developing non-local LES subgrid-scale models, starting from the kinetic transport. We employ a tempered Lévy-stable distribution to represent the source of turbulent effects at the kinetic level, and we rigorously show that the corresponding turbulence closure term emerges as the tempered fractional Laplacian, $(\varDelta +\lambda )^{\alpha } (\cdot )$ , for $\alpha \in (0,1)$ , $\alpha \neq \frac {1}{2}$ and $\lambda >0$ in the filtered Navier–Stokes equations. Moreover, we prove the frame invariant properties of the proposed model, complying with the subgrid-scale stresses. To characterize the optimum values of model parameters and infer the enhanced efficiency of the tempered fractional subgrid-scale model, we develop a robust algorithm, involving two-point structure functions and conventional correlation coefficients. In an a priori statistical study, we evaluate the capabilities of the developed model in fulfilling the closed essential requirements, obtained for a weaker sense of the ideal LES model (Meneveau, Phys. Fluids , vol. 6, issue 2, 1994, pp. 815–833). Finally, the model undergoes the a posteriori analysis to ensure the numerical stability and pragmatic efficiency of the model. 

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5. Sharp complexity asymptotics and topological trivialization for the ( p , k ) spiked tensor model

   https://doi.org/10.1063/5.0070300  

   Auffinger, Antonio ; Ben Arous, Gerard ; Li, Zhehua ( April 2022 , Journal of Mathematical Physics)

   Using precise random matrix theory tools and the Kac–Rice formula, we provide sharp O(1) asymptotics for the average number of deep minima of the ( p, k) spiked tensor model. These sharp estimates allow us to prove that, when the signal-to-noise ratio is large enough, the expected number of deep minima is asymptotically finite as N tends to infinity and to establish the occurrence of topological trivialization by showing that this number vanishes when the strength of the signal-to-noise ratio diverges. We also derive an explicit formula for the value of the absolute minimum (the limiting ground state energy) on the N-dimensional sphere, similar to the recent work of Jagannath, Lopatto, and Miolane [Ann. Appl. Probab. 4, 1910–1933 (2020)]. 

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